Question

How do decimals change when multiplied or divided by powers of ten greater than 2

Answer 1

When decimals are multiplied or divided by powers of ten, the decimal point shifts to the right or left, respectively, depending on whether you're multiplying or dividing.

1. **Multiplying by Powers of Ten:**
- When you multiply a decimal by \(10^n\) (where \(n\) is a positive integer), the decimal point shifts to the right \(n\) places.
- For example:
- \(0.45 \times 10^2 = 0.45 \times 100 = 45\) (the decimal point shifts 2 places to the right).
- \(3.12 \times 10^3 = 3.12 \times 1000 = 3120\) (the decimal point shifts 3 places to the right).

2. **Dividing by Powers of Ten:**
- When you divide a decimal by \(10^n\), the decimal point shifts to the left \(n\) places.
- For example:
- \(5.67 \div 10^2 = 5.67 \div 100 = 0.0567\) (the decimal point shifts 2 places to the left).
- \(8.4 \div 10^3 = 8.4 \div 1000 = 0.0084\) (the decimal point shifts 3 places to the left).

### Summary:
- **Multiply by \(10^n\)**: Shift the decimal point **to the right** \(n\) places.
- **Divide by \(10^n\)**: Shift the decimal point **to the left** \(n\) places.

This principle applies for any positive integer \(n\), including values greater than 2.